Let \(X\)  be a quasi-projective Calabi-Yau threefold over \(\mathbf {C} \) , so that \(X\)  is smooth and \(K_X\cong \mathcal {O} _X\) . Fix a curve class \(\beta \in H_2(X)\) . Let \(M=M^X_{\beta }\)  be the moduli space of Simpson semistable [1]}, pure, 1-dimensional sheaves \(\mathcal {F} \)  with proper support on \(X\)  with \(\text{ch}_2(\mathcal {F} ) = \beta ^{\vee }\)  and \(\chi (\mathcal {F} ) = 1\) . The genus 0 Gopakumar-Vafa invariants \(n^0_{\beta }(X)\)  are defined mathematically by Katz [2]}:
